Contents

Idea

One fundamental tool in a knot theorist’s toolbox is the knot diagram. Classically, a knot is an embedding of the circle S1S^1 into Euclidean space ℝ3\mathbb{R}^3, and knot diagrams arise by projecting back down to the planeℝ2\mathbb{R}^2 to obtain a 4-valent plane graph (referred to as the shadow of the knot), while keeping track of whether each vertex corresponds to an under-crossing or an over-crossing. Although the knot is traditionally seen as embedded in ℝ3\mathbb{R}^3, it could equally well be realized inside of a “thickened” sphereS2×[0,1]S^2 \times [0,1], with its shadow embedded in S2S^2. From that perspective, virtual knot theory generalizes classical knot theory by considering knots as embeddings of circles into thickened orientable surfaces X×[0,1]X \times [0,1] of arbitrary genus. Abstractly, the shadow of such a knot is a 4-valent graph embedded in XX (i.e., a topological map). However, if one tries to project the knot onto the plane, the corresponding diagram might contain crossings that do not represent places where the knot passes over/under itself inside the thickened surface, but rather are artifacts of the knot’s non-planar shadow. So, in a virtual knot diagram such crossings are explicitly indicated as “virtual”, using a distinct notation from that for under/overcrossings.

In particular, Lebed shows that the virtual braid group VBnVB_n is isomorphic to the group of endomorphisms End𝒞2br(V⊗n)End_{\mathcal{C}_{2br}}(V^{\otimes n}), where 𝒞2br\mathcal{C}_{2br} is the free symmetric monoidal category generated by a single braided object VV.

The intuition here is that the (symmetric) braiding of the ambient symmetric monoidal category represents virtual crossings “for free”, while the braiding σ\sigma on the object VV represents “real” over- and under-crossings. (Compare some remarks by John Baez, which are similar in spirit.)

Terminology

Warning: a virtual knot/link has a genus in the sense of the genus of the underlying thickened surface into which it embeds (or equivalently, the genus of its shadow as a topological map), but this is unrelated to the classical notion of knot genus, in the sense of the minimal genus of a Seifert surface whose boundary is the knot.

Victoria Lebed, Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles, Thèse, Université Paris Diderot, 2012. (pdf) Note that the title is in French (“Braided objects: a unifying study of algebraic structures and a categorification of virtual braids”) but the main text of the thesis is in English.

Victoria Lebed, Categorical Aspects of Virtuality and Self-Distributivity, Journal of Knot Theory and its Ramifications, 22 (2013), no. 9, 1350045, 32 pp. (doi) According to the author, arXiv:1206.3916 is “an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles”.

Last revised on December 22, 2016 at 10:04:07.
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